A003135 -id:A003135 - cp's OEIS frontend

A109095 Numbers N such that N! is the product of exactly two smaller factorials (larger than 1).

6, 10, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000, 25852016738884976640000

Offset: 1

Jud McCranie, Jun 19 2005

nonn

N = x! is considered to be a trivial solution because then N! = N*(N-1)! = x!*(N-1)!. Therefore every factorial appears in this sequence.
All terms except a(2) = 10 appear to be trivial solutions. (From Erdős's paper this is known as Surányi's conjecture.)
Habsieger established that the least nontrivial solution must have N > 10^3000. - M. F. Hasler, Jan 19 2023

			10! = 6! * 7!, so 10 is in the sequence.

Richard K. Guy, Unsolved Problems in Number Theory, B23 Equal products of factorials, Springer, Third Edition, 2004, p. 123.
Laurent Habsieger, Explicit bounds for the Diophantine equation A!B! = C!, Fibonacci Quarterly (2019), 57, 1.

Paul Erdős, Problems and results on number theoretic properties of consecutive integers and related questions, Proc. 5th Manitoba Conf. Numerical Math., Congress. Num. 16 (1975), 25-44.
Laurent Habsieger, Explicit Bounds For The Diophantine Equation A!B! = C!, arXiv:1903.08370 [math.NT], 2019.

Cf. A000142, A034878, A001013, A003135, A058295, A075082, A109096, A109097.

is_A109095(n) = my(m=1, f=n!); while(n-->m, while(n!2));
select(is_A109095, [0..777]) \\ M. F. Hasler, Jan 19 2023

Definition corrected by Jon E. Schoenfield, Jul 02 2010

More terms from M. F. Hasler, Jan 19 2023

A109096 Numbers n such that n! is the product of exactly three smaller factorials.

Original entry on oeis.org

4, 10, 12, 16, 36, 48, 144, 240, 576, 720, 1440, 2880, 4320, 10080, 14400, 17280, 30240, 80640, 86400

Offset: 1

Jud McCranie, Jun 19 2005

nonn

			144! = 3! * 4! * 143!, so 144 is in the sequence.

Cf. A034878, A001013, A003135, A058295, A075082, A109095, A109097.

Definition corrected by Jon E. Schoenfield, Jul 02 2010

A109097 Numbers n such that n! is the product of exactly four smaller factorials.

Original entry on oeis.org

8, 9, 24, 72, 96, 216, 288, 480, 864, 1152, 1440, 2880, 3456, 4320, 5760, 8640, 13824, 17280, 20160, 25920, 28800, 34560, 60480, 69120, 86400

Offset: 1

Jud McCranie, Jun 19 2005

nonn

			86400! = 3! * 5! * 5! * 86399!, so 86400 is in the sequence.

Cf. A034878, A001013, A003135, A058295, A075082, A109095, A109096.

Definition corrected by Jon E. Schoenfield, Jul 02 2010

A109098 Numbers n such that n! is the product of exactly 5 smaller factorials (greater than 1).

Original entry on oeis.org

16, 48, 144, 192, 432, 576, 960, 1296, 1728, 2304, 2880, 5184, 5760, 6912, 8640, 11520, 17280, 20736, 25920, 27648, 34560, 40320, 51840, 57600, 69120, 82944

Offset: 1

Jud McCranie, Jun 19 2005

nonn

			144! = 2! * 2! * 3! * 3! * 143!, so 144 is in the sequence.

Cf. A034878, A001013, A003135, A058295, A075082, A109095, A109096, A109097.

A109100 Numbers n such that n! is the product of exactly 2 smaller factorials (greater than 1).

Original entry on oeis.org

10, 16, 24, 48, 96, 144, 192, 288, 384, 720, 768, 864, 1440, 1536, 1728, 3072, 4320, 5184, 6144, 8640, 10368, 12288, 24576, 25920, 31104, 40320, 49152, 51840, 62208, 80640, 86400, 98304

Offset: 1

Jud McCranie, Jun 19 2005

nonn

			10! = 3! * 5! * 7! = 6! * 7!, so 10 is in the sequence.

Cf. A034878, A001013, A003135, A058295, A075082, A109095, A109096, A109097, A109098, A109100, A109101, A109102, A109103.

A109101 Numbers n such that n! is the product of exactly 3 smaller factorials (greater than 1).

Original entry on oeis.org

576, 1152, 2304, 2880, 3456, 4608, 5760, 6912, 9216, 11520, 18432, 20736, 23040, 36864, 41472, 46080, 73728, 92160

Offset: 1

Jud McCranie, Jun 19 2005

nonn

Cf. A034878, A001013, A003135, A058295, A075082, A109095, A109096, A109097, A109098, A109099, A109100, A109102, A109103.

A109102 Numbers n such that n! is the product of exactly 4 smaller factorials (greater than 1).

Original entry on oeis.org

13824, 17280, 27648, 34560, 55296, 82944

Offset: 1

Jud McCranie, Jun 19 2005

nonn

Cf. A034878, A001013, A003135, A058295, A075082, A109095, A109096, A109097, A109098, A109099, A109100, A109101, A109103.

A109099 Numbers n such that n! can be expressed as the product of smaller factorials > 2.

Original entry on oeis.org

6, 10, 24, 36, 120, 144, 216, 576, 720, 864, 1296, 2880, 3456, 4320, 5040, 5184, 7776, 13824, 14400, 17280, 20736, 25920, 30240, 31104, 40320, 46656, 69120, 82944, 86400

Offset: 1

Jud McCranie, Jun 19 2005

nonn

			86400! = 5! * 6! * 86399!, so 86400 is in the sequence.

Cf. A034878, A001013, A003135, A058295, A075082, A109095, A109096, A109097, A109098.

Definition corrected by Jon E. Schoenfield, Jul 02 2010

A109103 Smallest a(n) such that a(n)! can be expressed as the product of smaller factorials, using n distinct factorials greater than 1 (with repetitions allowed).

Original entry on oeis.org

4, 9, 288, 34560

Offset: 2

Jud McCranie, Jun 19 2005

nonn

			34560! = 2! * 3! * 4! * 5! * 34559!, using five different factorials, so a(5)=34560.

Cf. A034878, A001013, A003135, A058295, A075082, A109095, A109096, A109097, A109098, A109099, A109100, A109101, A109102.

A109104 Numbers n such that n! can be expressed as the product of the factorials of prime numbers, repetitions allowed.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 24, 32, 48, 72, 128, 192, 240, 384, 432, 480, 720, 864, 1152, 1440, 2592, 2880, 5040, 6144, 6912, 8192, 10080, 11520, 15360, 15552, 23040, 25920, 27648, 51840, 62208, 69120, 73728, 86400

Offset: 1

Jud McCranie, Jun 19 2005

nonn

			10! = 3! * 5! * 7!, so 10 is in the sequence.

Cf. A034878, A001013, A003135, A058295, A075082, A109095, A109096, A109097, A109098, A109099, A109100, A109101, A109102, A109103.

cp's OEIS Frontend

A109095 Numbers N such that N! is the product of exactly two smaller factorials (larger than 1).

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A109096 Numbers n such that n! is the product of exactly three smaller factorials.

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A109097 Numbers n such that n! is the product of exactly four smaller factorials.

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A109098 Numbers n such that n! is the product of exactly 5 smaller factorials (greater than 1).

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A109100 Numbers n such that n! is the product of exactly 2 smaller factorials (greater than 1).

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A109101 Numbers n such that n! is the product of exactly 3 smaller factorials (greater than 1).

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A109102 Numbers n such that n! is the product of exactly 4 smaller factorials (greater than 1).

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A109099 Numbers n such that n! can be expressed as the product of smaller factorials > 2.

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A109103 Smallest a(n) such that a(n)! can be expressed as the product of smaller factorials, using n distinct factorials greater than 1 (with repetitions allowed).

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A109104 Numbers n such that n! can be expressed as the product of the factorials of prime numbers, repetitions allowed.

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